Accidental crossings of eigenvalues in the one-dimensional complex PT-symmetric Scarf-II potential

•Mixing of two branches of discrete eigenvalues of PT-symmetric Scarf II.•Accidental crossings of energy eigenvalues in one dimension, yet no degeneracy.•Loss of diagonalizability of Hamiltonian at the points of level crossings.•Negative energy poles of transmission coefficients.•A rare property of Jacobi polynomials used and proved.

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: E±n(λ) with n=0,1,2…n=0,1,2… . Then if strength (λ  ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ=λ⁎λ=λ⁎; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ=λ⁎λ=λ⁎ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ=λcλ=λc. To re-emphasize, sharply at λ=λ⁎λ=λ⁎ and λcλc, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.